Given: ABC CD bisects AB CD AB Prove: ACD BCD. Statement 1. ABC CD bisects AB. Reasons. 1. Given
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1 Given: ABC CD bisects AB CD AB Prove: ACD BCD 1. ABC CD bisects AB CD AB 2. AD DB Side 3. CDA and CDB are right 4. CDA CDB Angle 5. CD CD Side 6. ACD BCD 2. A bisector cuts a segment into 2 parts. 3. lines form right. 4. All rt are. 5. Reflexive post. 6. SAS SAS #2 Given: ABC and DBE bisect each other. Prove: ABD CBD 1. ABC and DBE bisect each other. 2. AB BC Side BD BE Side 3. ABD and BEC are vertical 4. ABD BEC Angle 5. ABD CBD 2. A bisector cuts a segment into 2 parts. 3. Intersecting lines form vertical. 4. Vertical are. 5. SAS SAS
2 #3 Given: AB CD and BC DA DAB, ABC, BCD and CDA are rt Prove: ABC ADC 1. AB CD Side BC DA Side 2. DAB, ABC, BCD and CDA are rt 3. ABC ADC Angle 4. ABC ADC 2. Given 3. All rt are. 4. SAS SAS #4 Given: PQR RQS PQ QS Prove: PQR RQS 1. PQR RQS Angle PQ QS Side 2. RQ RQ Side 3. PQR RQS 2. Reflexive Post. 3. SAS SAS
3 #5 Given: AEB & CED intersect at E E is the midpoint AEB AC AE & BD BE Prove: AEC BED 1. AEB & CED intersect at E E is the midpoint AEB AC AE & BD BE 2. AEC and BED are vertical 3. AEC BED Angle 4. AE EB Side 5. A & B are rt. 6. A B Angle 7. AEC BED 2. Intersecting lines form vertical. 3. Vertical are. 4. A midpoint cut a segment into 2 parts 5. lines form right. 6. All rt are. 7. ASA ASA #6 Given: AEB bisects CED AC CED & BD CED Prove: EAC EBD 1. AEB bisects CED AC CED & BD CED 2. CE ED Side 3. ACE & EDB are rt 4. ACE EDB Angle 2. A bisector cuts an angle into 2 parts. 3. Lines form rt. 4. All rt are
4 5. AEC & DEB are vertical 6. AEC DEB Angle 7. EAC EBD #7 5. Intersect lines form vertical 6. Vertical are 7. ASA ASA Given: ABC is equilateral D midpoint of AB Prove: ACD BCD 1. ABC is equilateral D midpoint of AB 2. AC BC Side 3. AD DB Side 4. CD CD Side 5. ACD BCD 2. All sides of an equilateral are 3. A midpoint cuts a segment into 2 parts. 4. Reflexive Post 5. SSS SSS #8 Given: m A = 50, m B = 45, AB = 10cm, m D = 50 m E = 45 and DE = 10cm Prove: ABC DEF 1. m A = 50, m B = 45, AB = 10cm, m D = 50 m E = 45 and DE = 10cm 2. A = D Angle and B = E Angle AB = DE Side 3. ABC DEF 2. Transitive Prop 3. ASA ASA
5 #9 Given: GEH bisects DEF m D = m F Prove: GFE DEH 1. GEH bisects DEF m D = m F Angle 2. DE EF Side 3. 1 & 2 are vertical Angle 5. GFE DEH 2. Bisector cut a segment into 2 parts. 3. Intersect lines form vertical 4. Vertical are 5. ASA ASA #10 Given: PQ bisects RS at M R S Prove: RMQ SMP 1. PQ bisects RS at M R S Angle 2. RM MS Side 2. Bisector cut a segment into 2
6 3. 1 & 2 are vertical angles Angle 5. RMQ SMP parts 3. Intersect lines form vertical 4. Vertical are 5. ASA ASA #11 Given: DE DG EF GF Prove: DEF DFG 1. DE DG Side EF GF Side 2. DF DF Side 3. DEF DFG 2. Reflexive Post 3. SSS SSS #12 Given: KM bisects LKJ LK JK Prove: JKM LKM 1. KM bisects LKJ LK JK Side Angle 2. An bisectors cuts the into 2 parts
7 3. KM KM Side 4. JKM LKM 3. Reflexive Post 4. SAS SAS #13 Given:. PR QR P Q RS is a median Prove: PSR QSR 1. PR QR Side P Q Angle RS is a median Side 2. PS SQ 3. PSR QSR 2. A median cuts the side into 2 parts 3. SAS SAS #14 Given: EG is bisector EG is an altitude Prove: DEG GEF 1. EG is bisector EG is an altitude Angle 2. An bisector cuts an into 2 parts.
8 3. EG DF 4. 1 & 2 are rt Angle 6. GE GE Side 7. DEG GEF 3. An altitude form lines. 4. lines form right angles. 5. All right angles are 6. Reflexive Post 7. ASA ASA #15 Given: A and D are a rt AE DF AB CD Prove: EC FB 1. A and D are a rt AE DF Side AB CD 2. A D Angle 3. BC BC 4. AB + BC CD + BC or AC BD Side 5. AEC DFB 6. EC FB 2. All right angles are. 3. Reflexive Post. 4. Addition Prop. 5. SAS SAS 6. Corresponding parts of are. #16 Given: CA CB D midpoint of AB Prove: A B 1. CA CB Side D midpoint of AB
9 2. AD DB Side 3. CD CD Side 4. ADC DBC 5. A B 2. A midpoint cuts a segment into 2 parts 3. Reflexive Post 4. SSS SSS 5. Corresponding parts of are. #17 Given:. AB CD CAB ACD Prove: AD CB 1. AB CD Side CAB ACD Angle 2. AC AC Side 3. ACD ABC 4. AD CB 2. Reflexive Post 3. SAS SAS 4. Corresponding parts of are. #18 Given: AEB & CED bisect each Other Prove: C D 1. AEB & CED bisect each other 2. CE ED Side & AE EB Side 3. 1 and 2 are vertical 2. A bisector cuts segments into 2 parts. 3. Intersect lines form vertical
10 Angle 5. AEC DEB 6. C D 4. Vertical are 5. SAS SAS 6. Corresponding parts of are #19 Given: KLM & NML are rt KL NM Prove: K N 1. KLM & NML are rt KL NM Side 2. KLM NML Angle 3. LM LM Side 4. KLM LNM 5. K N 2. All rt are 3. Reflexive Post. 4. SAS SAS 5. Corresponding parts of are. #20 Given: AB BC CD PA PD & PB PC Prove: a) APB DPC b) APC DPB 1. AB BC CD Side PA PD Side & PB PC Side 2. ABP CDP 3. APB DPC 2. SSS SSS 3. Corresponding parts of are.
11 4. BPC BPC 5. APB + BPC DPC + BPC or APC DPB 4. Reflexive Post. 5. Addition Prop. #21 Given: PM is Altitude PM is median Prove: a) LNP is isosceles b) PM is bisector 1. PM is Altitude & PM is median 2. PM LN 3. 1 and 2 are rt LM MN 6. PM PM 7. LMP PMN 8. PL PN 9. LNP is isosceles 10. LPN MPN 11. PM is bisector 2. An altitude form lines. 3. lines form right angles. 4. All right angles are 5. A median cuts the side into 2 parts 6. Reflexive Post. 7. SAS SAS 8. Corresponding parts of are. 9. An Isosceles is a with2 sides 10.Corresponding parts of are. 11. A bisector cuts an into 2 parts #22
12 Given: CA CB Prove: CAD CBE 1. CA CB & 2 are supplementary 3 & 4 are supplementary or CAD CBE 2. If 2 sides are then the opposite are. 3. Supplementary are form by a linear pair. 4. Supplement of are. #23 Given: AB CB & AD CD Prove: BAD BCD 1. AB CB & AD CD or BAD BCD 2. If 2 sides are then the opposite are. 3. Addition Post. #24
13 Given: ΔABC ΔDEF M is midpoint of AB N is midpoint DE Prove: ΔAMC ΔDNF 1. ΔABC ΔDEF 2. M is midpoint of AB N is midpoint DE 3. D A Angle and DF AC Side 4. AM MB and DN NE Side 5. ΔAMC ΔDNF 2. Given 3. Corresponding parts of Δ are 4. A midpoint cuts a segment into 2 parts 5. SAS SAS #25 Given: ΔABC ΔDEF CG bisects ACB FH bisects DFE Prove: CG FH 1. ΔABC ΔDEF CG bisects ACB FH bisects DFE
14 #26 Given: ΔAME ΔBMF DE CF Prove: AD BC 1. ΔAME ΔBMF DE CF 2. EM MF AM MB Side 1 2 Angle 3. DE + EM CF + MF or DM MC Side 4. ΔADM ΔBCM 5. AD BC 2. Corresponding parts of Δ are 3. Addition Post. 4. SAS SAS 5. Corresponding parts of Δ are Given: AEC & DEB bisect each other Prove: E is midpoint of FEG 1. AEC & DEB bisect each other
15 2. DE BE Side and AE EC Side 3. AEB & DEC are vertical 4. AEB DEC Angle 5. ΔAEB ΔDEC 6. D B 7. 1 & 2 are vertical angles ΔGEB ΔDEF 10. GE FE 11. E is midpoint of FEG 2. A bisector cuts a segment into 2 parts. 3. Intersecting lines form vertical 4. Vertical are. 5. SAS SAS 6. Corresponding parts of Δ are 7. Intersecting lines form vertical 8. Vertical are. 9. ASA ASA 10. Corresponding parts of Δ are 11. A midpoint divides a segment into 2 parts. #28 Given: BC BA BD bisects CBA Prove: DB bisects CDA
16 1. BC BA Side BD bisects CBA Angle 3. BD BD Side 4. ΔABD ΔBCD DB bisects CDA 2. A bisector cuts an angle into 2 parts. 3. Reflexive Post. 4. SAS SAS 5. Corresponding parts of Δ are 6. A angle bisector cuts an angle into 2 parts. #29 Given: AE FB DA CB A and B are Rt. Prove: ADF CBE DF CE 1. AE FB DA CB Side A and B are Rt. 2. EF EF 3. AE + EF FB + EF or AF EB Side 2. Reflexive Post 3. Addition Property
17 4. A B Angle 5. ADF CBE 6. DF CE 4. All rt. are. 5. SAS SAS 6. Corresponding parts of Δ are #30 Given: SPR SQT PR QT Prove: SRQ STP R T 1. SPR SQT Side PR QT 2. S S Angle 3. SPR PR SQT QT or SR ST 4. SRQ STP 5. R T #31 Side 2. Reflexive Post 3. Subtraction Property 4. SAS SAS 5. Corresponding parts of Δ are Given: DA CB DA AB & CB AB Prove: DAB CBA AC BD 1. DA CB Side DA AB & CB AB 2. DAB and CBA are rt 3. DAB CBA Angle 4. AB AB Side 5. DAB CBA 6. AC BD 2. lines form rt. 3. All rt are. 4. Reflexive post. 5. SAS SAS 6. Corresponding parts of Δ are.
18 #32 Given: BAE CBF BCE CDF AB CD Prove: AE BF E F 1. BAE CBF Angle BCE CDF Angle AB CD 2. BC BC 3. AB + BC CD + BC or AC BD Side 4. AEC BDF 5. AE BF E F 2. Reflexive Post. 3. Addition Property. 4. ASA ASA 5. Corresponding parts of Δ are. #33 Given: TM TN M is midpoint TR N is midpoint TS Prove: RN SM
19 1. TM TN Side M is midpoint TR N is midpoint TS 2. T T Angle 3. RM is ½ of TR NS is ½ of TS 4. RM NS 5. TM + RM TN + NS or RT TS Side 6. RTN MTS 7. RN SM 2. Reflexive Post. 3. A midpoint cuts a segment in. 4. ½ of parts are. 5. Addition Property 6. SAS SAS 7. Corresponding parts of Δ are. #34 Given: AD CE & DB EB Prove: ADC CEA 1. AD CE & DB EB Side
20 2. B B Angle 3. AD + DB CE + EB or AB BC Side 4. ABE BCD & 3 are supplementary 2 & 4 are supplementary or ADC CEA 2. Reflexive Post 3. Addition Post. 4. SAS SAS 5. Corresponding parts of Δ are. 6. A st. line forms supplementary. 7. Supplements of are. #35 Given: AE BF & AB CD ABF is the suppl. of A Prove: AEC BFD 1. AE BF Side & AB CD ABF is the suppl. of A
21 2. A 1 Angle 3. BC BC 4. AB + BC CD + BC or AC BD Side 5. AEC BFD 2. Supplements of are. 3. Reflexive Post. 4. Addition Property. 5. SAS SAS #36 Given: AB CB BD bisects ABC Prove: AE CE 1. AB CB Side BD bisects ABC Angle 3. BE BE Side 4. BEC BEA 5. AE CE 2. A bisector cuts an into 2 parts. 3. Reflexive Post. 4. SAS SAS 5. Corresponding parts of Δ are #37 Given: PB PC Prove: ABP DCP 1. PB PC
22 & ABP are supplementary 2 & DCP are supplementary 4. ABP DCP 2. opposite sides are. 3. Supplementay are formed by a linear pair. 4. Supplements of are. #38 Given: AC and BD are bisectors of each other. Prove: AB BC CD DA 1. AC and BD are bisectors of each other 2. 1, 2, 3 and 4 are rt Angle 4. AE EC and BE DE 2 sides 5. ABE BEC DEC AED 6. AB BC CD DA 2. lines form rt. 3. All rt are. 4. A bisector cuts a segment into 2 parts. 5. SAS SAS 6. Corresponding parts of Δ are #39 Given: AEFB, 1 2 CE DF, AE BF Prove: AFD BEC
23 1. AEFB, 1 2 Angle CE DF Side, AE BF 2. EF EF 3. AE + EF BF + EF or AF EB Side 4. AFD BEC 2. Reflexive Post. 3. Addition Property 4. SAS SAS #40 Given: SX SY, XR YT Prove: RSY TSX 1. SX SY Side, XR YT 2. SX + XR SY + YT or SR ST Side 3. S S Angle 4. RSY TSX 2. Addition Post. 3. Reflexive Post. 4. SAS SAS #41 Given: DA CB DA AB, CB AB Prove: DAB CBA
24 1. DA CB Side DA AB, CB AB 2. DAB and CBA are rt. 3. DAB CBA Angle 4. AB AB Side 5. DAB CBA 2. lines form rt 3. All rt. are 4. Reflexive Post. 5. SAS SAS #42 Given: AF EC 1 2, 3 4 Prove: ABE CDF 1. AF EC 1 2, 3 4 Angle 2. DFC BEA Angle 3. EF EF 4. AF + EF EC + EF or AE FC Side 5. ABE CDF 2. Supplements of are 3. Reflexive post. 4. Addition Post. 5. AAS AAS #43
25 Given: AB BF, CD BF 1 2, BD FE Prove: ABE CDF 1. AB BF, CD BF 1 2 Side, BD FE 2. B and CDF are rt. 3. B CDF Angle 4. DE DE 5. BD + DE FE + DE or BE DF Side 6. ABE CDF 2. lines form rt. 3. All rt. are 4. Reflexive Post. 5. Addition Post. 6, ASA ASA #44 Given: BAC BCA CD bisects BCA AE bisects BAC Prove: ADC CEA 1. BAC BCA Angle CD bisects BCA AE bisects BAC 2. ECA ½ BAC and DCA ½ BCA 3. ECA DCA Angle 4. AC AC Side 5. ADC CEA 2. bisector cuts an in ½ 3. ½ of are 4. Reflexive post. 5. ASA ASA
26 #45 Given: TR TS, MR NS Prove: RTN STM 1. TR TS Side, MR NS 2, TR MR TS NS or TM TN Side 3. T T Angle 4. RTN STM #46 2. Subtraction Post. 3. Reflexive Post. 4. ASA ASA Given: CEA CDB, ABC AD and BE intersect at P PAB PBA Prove: PE PD 1. CEA CDB, ABC AD and BE intersect at P PAB PBA 2.
27 #47 Given: AB AD and BC DC Prove: AB AD and BC DC 2. AC AC 3. ABC ADC 4. AE AE 5. BAE DAE 6. ABE ADE Reflexive Post. 3. SSS SSS 4. Reflexive Post. 5. Corresponding parts of Δ are. 6. SAS SAS 7. Corresponding parts of Δ are. #48 Given: BD is both median and altitude to AC Prove: BA BC 1. BD is both median and altitude to AC 2. AD CD Side 3. ADB and CDB are rt. 4. ADB CDB Angle 5. BD BD Side 6. ABD CBD 2. A median cuts a segment into 2 parts 3. Lines form rt. 4. All rt. are 5. Reflexive Post.
28 7. BA BC 6. SAS SAS 7. Corresponding parts of Δ are. #49 Given: CDE CED and AD EB Prove: ACC BCE 1. CDE CED and AD EB Side 2. CDA CEB Angle 3. CD CE Side 4. ADC BEC 5. ACD BCE 2. Supplements of are. 3. Sides opp. in a are 4. SAS SAS 5. Corresponding parts of Δ are. #50 Given: Isosceles triangle CAT CT AT and ST bisects CTA Prove: SCA SAC 1. Isosceles triangle CAT CT AT Side and ST bisects CTA 2. CTS ATS Angle 3. ST ST Side 4. CST AST 2. An bisector cuts an into 2 parts 3. Reflexive Post. 4. SAS SAS
29 5. CS AS 6. SCA SAC 5. Corresponding parts of Δ are. 6. opp. sides in a are #51 Given: 1 2 DB AC Prove: ABD CBD and DB AC 2. DBA and DBC are rt. 3. DBA DBC Angle 4. DAB DCA Angle 5. DB DB Side 6. ABD CBD 2. lines form rt. 3. All rt. are 4. Supplements of are 5. Reflexive Post. 6. AAS AAS #52 Given: P S R is midpoint of PS Given: PQR STR 1. P S Angle R is midpoint of PS 2. PR RS Side 3. QRP and TRS are vertical 2. A midpoint cuts a segment into 2 parts 3. Intersecting lines form vert.
30 4. QRP TRS Angle 5. PQR STR 4. Vertical are 5. ASA ASA #53 Given: FG DE G is midpoint of DE Given: DFG EFG 1. FG DE G is midpoint of DE 2. FGD and FGE are rt. 3. FGD FGE Angle 4. FG FG Side 5. DG GE Side 6. DFG EFG 2. lines form rt. 3. All rt. are 4. Reflexive Post. 5. A midpoint cuts a segment into 2 parts. 6. SAS SAS #54 Given: AC CB D is midpoint of AB Prove: ACD BCD 1. AC CB Side D is midpoint of AB
31 2. AD DB Side 3. CD CD Side 4. ACD BCD 2. A midpoint cuts a segment into 2 parts. 3. Reflexive Post. 4. SSS SSS #55 Given: PT bisects QS PQ QS and TS QS Prove: PQR RST 1. PT bisects QS PQ QS and TS QS 2. QR RS Side 3. PRQ and TRS are vertical 4. PRQ TRS Angle 5. Q and S are rt. 6. Q S Angle 7. PQR RST 2. A bisector cuts a segment into 2 parts. 3. Intersecting lines form vert. 4. All vert. are 5. lines form rt. 6. All rt. are 7. ASA ASA #56 Given: AB ED and FE CB FE AD and CB AD Prove: AEF CBD 1. AB ED and FE CB Side
32 FE AD and CB AD 2. BE BE 3. AB + BE ED + BE or AE DB Side 4. AEF and DBF are rt. 5. AEF DBF Angle 6. AEF CBD #57 2. Reflexive Post. 3. Addition Post. 4. lines form rt. 5. All rt. are 6. SAS SAS Given: SM is bisector of LP RM MQ a b Prove: RLM QPM 1. SM is bisector of LP RM MQ Side a b 2. SML and SMP are rt Angle 4. LM PM Side 5. RLM QPM 2. lines form rt. 3. Complements of are 4. A bisector cuts a segment into 2 parts. 5. SAS SAS #59 Given: AC BC CD AB Prove: ACD BCD
33 1. AC BC CD AB 2. CDA and CDB are rt. 3. CDA CDB 4. CD CD 5. ACD BCD 2. lines form rt. 3. All rt. are 4. Reflexive Post. 5. SAS SAS #60 Given: FQ bisects AS A S Prove: FAT QST 1. FQ bisects AS A S Angle 2. AT ST Side 3. ATF & STQ are vertical 4. ATF STQ Angle 5. FAT QST 2. A bisector cuts a segment into 2 parts. 3. Intersecting lines form vert. 4. All vert. are 5. ASA ASA #61 Given: A D and BCA FED AE CD AEF BCD Prove: ABC DFE 1. A D Angle and BCA FED Angle
34 AE CD and AEF BCD 2. EC EC 3. AE + EC CD + EC or AC DE Side 4. ABC DFE 2. Reflexive Post. 3. Addition Post. 4. ASA ASA #62 Given: SU QR, PS RT TSU QRP Prove: PQR STU Q U 1. SU QR, PS RT TSU QRP 2. SR SR 3. PS + SR = RT + SR or PR TS 4. PQR STU 5. Q U 2. Reflexive Post. 3. Addition Post 4. SAS SAS 5. Corresponding parts of Δ are. #63
35 Given: M D ME HD THE SEM Prove: MTH DSE 1. M D Angle, ME HD THE SEM 2. HE HE 3. ME HE HD - HE or MH DE Side 4. THM SED Angle 5. MTH DSE 2. Reflexive post. 3. Subtraction Post. 4. Supplements of are 5. ASA ASA #64 Given; SQ bisects PSR P R Prove: PQS QSR 1. SQ bisects PSR P R Angle 2. PSQ RSQ Angle 3. SQ SQ Side 4. PQS QSR 2. an bisectors cuts an into 2 parts. 3. Reflexive Post 4. AAS AAS
36 #65 Given: PQ QS and TS QS R midpoint of QS Prove: P T 1. PQ QS and TS QS R midpoint of QS 2. Q and S are rt. 3. Q S Angle 4. PRQ and TRS are vertical 5. PRQ TRS Angle 6. QR SQ Side 2. lines form rt. 3. All rt. are 4. Intersecting lines form vert. 5. All vert. are 6. A midpoint cuts a segment into 2
37 7. PQR TSR 8. P T parts. 7. ASA ASA 8. Corresponding parts of Δ are. #66 Given: CB FB, BT BV DV TS, DC FS Prove: D S 1. CB FB, BT BV DV TS, DC FS Side 2. BTV BVT Angle 3. CB + BT FB + BV or CT FV Side 4. VT VT 5. DV + VT TS + VT or DT SV Side 6. DCT SVF 7. D S 2. opp. sides in a are 3. Addition Post 4. Reflexive Post. 5. Addition Post 6. SAS SAS 7. Corresponding parts of Δ are.
38 #67 Given: PQ DE and PB AE QA PE and DB PE Prove: D Q 1. PQ DE Hyp and PB AE QA PE and DB PE 2. AB AB 3. PB AB = AE AB or PA EB Leg 4. QAP and DBA are rt. 2. Reflexive post. 3. Subtraction Post. 4. lines form rt.
39 5. QAP DBA 6. PAQ EBD 7. D Q #68 5. All rt. are 6. HL HL 7. Corresponding parts of Δ are. Given: TS TR P Q Prove: PS QR 1. TS TR Side P Q Angle 2. PTS and QTR are vertical 3. PTS QTR Angle 4. PTS QTR 5. PS QR 2. Intersecting lines form vert. 3. All vert. are 4. AAS AAS 5. Corresponding parts of Δ are. #69 Given: HY and EV bisect each other Prove: HE VY 1. HY and EV bisect each other 2. HA YA Side and EA VA Side 3. HAE and YAV are vertical 4. HAE YAV Angle 5. HAE YAV 6. HE VY 2. A bisector cuts a segment into 2 parts. 3. Intersecting lines form vert. 4. All vert. are 5. SAS SAS 6. Corresponding parts of Δ are.
40 #70 Given: E D and A C B is the midpoint of AC Prove: EA DC 1. E D Angle and A C Angle B is the midpoint of AC 2. EA DC Side 3. ABE CBE 4. EA DC 2. A midpoint cuts a segment into 2 parts. 3. AAS AAS 4. Corresponding parts of Δ are. #71 Given: E is midpoint of AB DA AB and CB AB 1 2 Prove: AD CB 1. E is midpoint of AB DA AB and CB AB AE EB Side 3. DE CE Side 2. A midpoint cuts a segment into 2 parts. 3. opp. sides in a are
41 4. ADE BCD 5. AD CB 4. HL HL 5. Corresponding parts of Δ are.
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